958 Continuous-Time Stochastic Volatility Models
SV4 is also called the constant elasticity of variance process and has been used,
for example, by Jones (2003) and Aït-Sahalia and Kimmel (2007). The functional
form of the diffusion component implies that the model can easily accommodate
the level effect. The conditional mean of SV4 generalizes those of SV1 and SV2,
which are obtained by settingγto zero and 0.5, respectively. However, the con-
ditional variance does not have an analytical form and can only be derived using
approximations.
SV5 is by far the most popular specification of discrete-time stochastic volatil-
ity models as it produces Gaussian log-volatilities (for example, Jacquier, Polson
and Rossi, 1994; Kim, Shephard and Chib, 1998). In continuous-time settings, it
has been used by Wiggins (1987) for the pricing of equity options, by Melino and
Turnbull (1990) for the pricing of currency options, and by Detemple and Osakwe
(2000) and Psychoyios, Dotsis and Markellos (2007) for the valuation of volatility
options. Andersen, Benzoni and Lund (2002) and Chernovet al.(2003) estimated
the model empirically. From the volatility levels process we can deduce, first, that
the model accounts for the level effect of volatility and, second, that mean rever-
sion depends on the level ofVt, that is, the largerVt, the larger the mean reversion
of the process. The first two conditional central moments are given by:
Et(
Vt)
=Vtexp(−k(T−t))exp((
1 −e−λt)
μ+
σ^2
4 λ(
1 −e−^2 λt))
(19.8)Vart(
Vt)
=Vt2 exp(−k(T−t))exp(
2(
1 −e−k(T−t))
θ+
σ^2
2 k(
1 −e−k(T−t)))×(
exp(
σ^2
2 k(
1 −e−^2 k(T−t)))
− 1). (19.9)
19.3 Inference in stochastic volatility models
Although stochastic volatility models are built in continuous time, empirical data
are only observed at discrete time intervals. Inference in stochastic volatility models
is a difficult task because the likelihood function is typically not available in a
tractable form. The Gaussian QML approach of Harvey, Ruiz and Shephard (1994)
is appealing because of its simplicity, but many studies have shown that the method
fails because volatility models are highly non-Gaussian.
Suppose thaty=
{
y 1 ,...,yT}
is a discrete time series of observed data,yt=
log(St)−log(St− 1 ),t =1,...,T, andV =
{
V 1 ,...,VT}
is the vector of latent
stochastic volatility. We assume thatyis stationary and thatxt=(yt,Vt)forms a
Markov system. The systemxtcan be treated as either fully or partially observed. If
the former, latent volatility may be extracted from the derivatives markets, whereas,
under the latter, volatility is treated as a latent variable that has to be integrated
out of the likelihood function.